particle moving on a cardioid at constant frequency

This is another elementary example11C.F. particle moving on the astroid at constant frequency about particle kinematics. In this case we will use polar coordinates. Let us consider the cardioidMathworldPlanetmath 22the locus of the points of the plane described by a circle (or disc) boundary point which it is rolling over another one with the same radius R.

33indeed the native polar equation of the cardioid is r=2R(1+cosθ),θ=ωt.

with R,ω>0 given constants and t[0,) means time parameter. The position vector of a particle, respect to an orthonormal reference basis {𝐫^,θ^}, moving on the cardioid is


and its velocity 44in polar coordinates we have 𝐫˙=r˙𝐫^+rθ˙θ^, because the base vectors 𝐫^,θ^ are changing on direction and sense according the formulas d𝐫^dθ=θ^,dθ^dθ=-𝐫^. We are using the chain ruleMathworldPlanetmath with θ˙=ω. Overdot denotes time differentiationMathworldPlanetmath everywhere.


Therefore the speed is


and the tangent vectorMathworldPlanetmathPlanetmath


Next we use the formula


and by using the time derivative of base vectors


getting the equation

Title particle moving on a cardioid at constant frequency
Canonical name ParticleMovingOnACardioidAtConstantFrequency
Date of creation 2013-03-22 17:14:20
Last modified on 2013-03-22 17:14:20
Owner perucho (2192)
Last modified by perucho (2192)
Numerical id 6
Author perucho (2192)
Entry type Topic
Classification msc 70B05